4.12 Convergence of Taylor series.

Taylor’s theorem shows that the Taylor series associated with $f\,$ converges to $f(x)\,$ if the remainder term has $R_{n}(x)\rightarrow 0\,$ as $n\rightarrow\infty\,$. There are formulæ for the remainder term which allow one to check whether this condition holds in cases of interest. However, the binomial series

does not converge for $|x|>1\,$, so we cannot hope to represent all functions by Maclaurin series that converge everywhere. This topic will be considered in MATH113 and in second-year analysis courses.